📚 Exact Values of Trigonometric Ratios for 150° and Related Problems | 150° 的三角函数精确值及相关题目
In IGCSE Edexcel Mathematics, you are expected to know exact trigonometric values for special angles. The angle 150° is one of the most important obtuse angles because it is linked directly to 30° and appears frequently in non-calculator papers.
在 Edexcel IGCSE 数学中，你被要求牢记特殊角的精确三角函数值。150° 是最重要的钝角之一，因为它与 30° 紧密相关，并且在非计算器试卷中频繁出现。
1. What Does 150° Mean in Trigonometry? | 150° 在三角学中的意义
An angle of 150° is an obtuse angle because it is greater than 90° and less than 180°. In a right-angled triangle, angles are always acute, so we need a different way to think about sine, cosine and tangent for 150°.
150° 是大于 90° 且小于 180° 的角，所以它是钝角。在直角三角形中，角总是锐角，因此我们需要用另一种方式来理解 150° 的正弦、余弦和正切。
On the unit circle, we draw an angle anticlockwise from the positive x-axis. The point where the terminal side meets the circle has coordinates related to cos 150° and sin 150°.
在单位圆上，我们从 x 轴正方向开始逆时针旋转画出这个角。终边与单位圆交点的坐标就与 cos 150° 和 sin 150° 有关。
The angle 150° is in the second quadrant. In this quadrant, sine is positive, cosine is negative, and tangent is negative.
150° 位于第二象限。在第二象限中，正弦为正，余弦为负，正切为负。
2. The Reference Angle: Why 150° Connects to 30° | 参考角：为什么 150° 与 30° 相关
The reference angle is the acute angle made between the terminal side of a given angle and the x-axis. For 150°, the terminal side is 180° − 150° = 30° away from the x-axis.
参考角是指给定角的终边与 x 轴之间形成的锐角。对于 150°，终边距离 x 轴为 180° − 150° = 30°。
Therefore every trigonometric ratio of 150° has the same magnitude as the corresponding ratio for 30°. We only need to decide whether the sign is positive or negative.
因此，150° 的每一个三角函数值与 30° 对应值的绝对值相同。我们只需要判断正负号。
Reference angle of 150° = 30°
150° 的参考角 = 30°
3. Sin 150°: The Exact Value | sin 150° 的精确值
From the graph of y = sin θ or the unit circle, we know that sin 150° = sin 30°. This is because sine is positive in the second quadrant and symmetric about 90°.
从 y = sin θ 的图像或单位圆可知，sin 150° = sin 30°。这是因为正弦在第二象限为正，并且关于 90° 对称。
Since the exact value of sin 30° is ½, we obtain the exact value of sin 150°.
因为 sin 30° 的精确值是 ½，所以我们得到 sin 150° 的精确值。
sin 150° = ½
4. Cos 150°: The Exact Value | cos 150° 的精确值
Cos 150° has the same magnitude as cos 30°, but cosine is negative in the second quadrant.
cos 150° 与 cos 30° 的绝对值相同，但余弦在第二象限为负。
The exact value of cos 30° is √3/2. Therefore the exact value of cos 150° is −√3/2.
cos 30° 的精确值是 √3/2。因此，cos 150° 的精确值是 −√3/2。
cos 150° = −√3/2
5. Tan 150°: The Exact Value | tan 150° 的精确值
We can use the identity tan θ = sin θ / cos θ. For θ = 150°, this gives:
我们可以使用恒等式 tan θ = sin θ / cos θ。当 θ = 150° 时，得到：
tan 150° = (½) / (−√3/2) = −1/√3 = −√3/3
Because sine is positive and cosine is negative, tangent is negative. The magnitude is equal to tan 30° = 1/√3.
因为正弦为正、余弦为负，所以正切为负。其绝对值等于 tan 30° = 1/√3。
tan 150° = −√3/3
6. The Exact-Values Table for Common Angles | 常见角度的准确值表
The table below summarises the exact values you should memorise for angles between 0° and 180°.
下表总结了你在 0° 到 180° 之间应该牢记的精确值。
| Angle θ | sin θ | cos θ | tan θ |
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
| 120° | √3/2 | −½ | −√3 |
| 150° | ½ | −√3/2 | −√3/3 |
| 180° | 0 | −1 | 0 |
7. Coordinates on the Unit Circle | 单位圆上的坐标
On the unit circle, the point at angle θ has coordinates (cos θ, sin θ). Therefore the point at angle 150° has coordinates (−√3/2, ½).
在单位圆上，角度 θ 对应的点坐标为 (cos θ, sin θ)。因此，150° 对应的点坐标为 (−√3/2, ½)。
This coordinate appears in vector questions, rotation questions, and problems involving bearings or regular pentagons.
这个坐标会出现在向量题、旋转题，以及涉及方位角或正五边形的题目中。
If you are asked to write the coordinates of a point on a circle of radius r, multiply the unit-circle coordinates by r.
如果题目要求半径为 r 的圆上一点的坐标，则将单位圆坐标乘以 r。
P = (r cos 150°, r sin 150°) = (−r√3/2, r/2)
8. Solving Trigonometric Equations with 150° | 用 150° 解三角方程
Exact values allow us to solve trigonometric equations without a calculator. For example, solve sin θ = ½ for 0° ≤ θ ≤ 180°.
精确值让我们可以在不使用计算器的情况下解三角方程。例如，解 sin θ = ½，其中 0° ≤ θ ≤ 180°。
Since sin 30° = ½ and sine is also positive in the second quadrant, the second solution is 180° − 30° = 150°.
因为 sin 30° = ½，且正弦在第二象限也为正，所以第二个解是 180° − 30° = 150°。
θ = 30° or θ = 150°
For cos θ = −√3/2 in the same interval, the only solution is θ = 150°, because cosine is negative in the second quadrant and equals −√3/2 only when θ = 150° between 0° and 180°.
对于 cos θ = −√3/2，在同一区间内唯一的解是 θ = 150°，因为在 0° 到 180° 之间，余弦为负且等于 −√3/2 只在 θ = 150° 时成立。
9. 150° in the Sine Rule | 正弦定理中的 150°
The sine rule states that in any triangle, a/sin A = b/sin B = c/sin C. A triangle can contain an obtuse angle, and 150° is an important example.
正弦定理指出，在任意三角形中，a/sin A = b/sin B = c/sin C。三角形可以包含钝角，150° 就是一个重要例子。
Because sin 150° = ½, the sine rule works normally with obtuse angles. However, when finding an angle using the inverse sine function, a calculator will give an acute angle, so you must check whether the obtuse angle is also valid.
因为 sin 150° = ½，正弦定理可以正常处理钝角。然而，使用反正弦函数求角时，计算器只给出锐角，所以你必须判断钝角是否也符合条件。
For example, if sin B = 0.5 and 0° < B < 180°, then B = 30° or B = 150°. The correct choice depends on the other angles and the triangle's shape.
例如，如果 sin B = 0.5 且 0° < B < 180°，那么 B = 30° 或 B = 150°。具体选择取决于其他角以及三角形的形状。
10. Common Exam Mistakes and How to Avoid Them | 常见考试错误与避坑方法
One common mistake is writing cos 150° as positive √3/2. Always identify the quadrant first: 150° is in the second quadrant, where cosine is negative.
一个常见错误是把 cos 150° 写成正的 √3/2。务必先判断象限：150° 在第二象限，余弦为负。
Another mistake is forgetting that tan 150° can be written in two forms: −1/√3 and −√3/3. Both are acceptable in IGCSE exams, but you must not write it as positive.
另一个错误是忘记 tan 150° 可以写成两种形式：−1/√3 或 −√3/3。在 IGCSE 考试中两种都可以接受，但不能写成正值。
When solving sin θ = ½, many students only write θ = 30°. Remember that between 0° and 180° there are two solutions: 30° and 150°.
当解 sin θ = ½ 时，许多学生只写 θ = 30°。请记住，在 0° 到 180° 之间有两个解：30° 和 150°。
Finally, learn the exact-value table by heart. In non-calculator papers, there is no way to get the exact value of 150° unless you know the 30° ratios and the quadrant signs.
最后，请牢记精确值表。在非计算器试卷中，除非你知道 30° 的比值和象限符号，否则无法得到 150° 的精确值。
11. Exam-Style Practice Questions | 考试风格练习题
Question 1: Write down the exact value of cos 150°. Answer: −√3/2.
第 1 题：写出 cos 150° 的精确值。答案：−√3/2。
Question 2: Solve 2 sin θ = 1 for 0° ≤ θ ≤ 180°. Answer: Divide by 2 to get sin θ = ½, so θ = 30° or 150°.
第 2 题：解 2 sin θ = 1，其中 0° ≤ θ ≤ 180°。答案：两边除以 2，得 sin θ = ½，所以 θ = 30° 或 150°。
Question 3: A point P lies on a circle of radius 4 with centre O at the origin. OP makes an angle of 150° with the positive x-axis. Find the coordinates of P.
第 3 题：点 P 在以原点 O 为圆心、半径为 4 的圆上，OP 与 x 轴正方向成 150° 角。求点 P 的坐标。
Solution: x = 4 cos 150° = −2√3; y = 4 sin 150° = 2. So P = (−2√3, 2).
解答：x = 4 cos 150° = −2√3；y = 4 sin 150° = 2。所以 P = (−2√3, 2)。
Question 4: In triangle ABC, angle A = 150°, side a = 6 cm and side b = 2 cm. Find angle B if possible.
第 4 题：在三角形 ABC 中，角 A = 150°，边 a = 6 cm，边 b = 2 cm。如果可能，求角 B。
Using the sine rule: 6/sin 150° = 2/sin B, so 6/0.5 = 2/sin B, giving sin B = 1/6. Since B must be acute, B = sin⁻¹(1/6), not 150°.
使用正弦定理：6/sin 150° = 2/sin B，所以 6/0.5 = 2/sin B，得到 sin B = 1/6。因为 B 必须是锐角，所以 B = sin⁻¹(1/6)，而不是 150°。
12. Summary | 总结
Angle 150° is a second-quadrant angle with reference angle 30°. Its exact trig values are: sin 150° = ½, cos 150° = −√3/2, tan 150° = −√3/3.
150° 是第二象限角，参考角为 30°。它的精确三角函数值为：sin 150° = ½，cos 150° = −√3/2，tan 150° = −√3/3。
Always remember the quadrant signs: sine positive, cosine negative, tangent negative in the second quadrant.
始终牢记象限符号：第二象限中正弦为正，余弦为负，正切为负。
Mastering 150° and its relationship with 30° will help you solve exact-value questions, unit circle questions, trigonometric equations and sine-rule problems quickly and accurately in the Edexcel IGCSE exam.
掌握 150° 与 30° 的关系，将帮助你在 Edexcel IGCSE 考试中快速准确地解决精确值题、单位圆题、三角方程题和正弦定理题。
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